Animated Solution for Physics - Thermodynamics: 1 mol of a gas with $\gamma =\frac{7}{5}$ is mixed with 1 mol of a gas with $\gamma =\frac{5}{3}$, then the value of $\gamma$ for the resulting mixture is

Imagine you have two different gases in separate containers. One is a diatomic gas with $\gamma =\frac{7}{5}$ (like Oxygen or Nitrogen), and the other is a monatomic gas with $\gamma =\frac{5}{3}$ (like Helium or Argon). What happens when you mix them together? Does the new mixture behave like a monatomic gas, a diatomic gas, or something entirely different?

To find the effective $\gamma$ of the mixture, we cannot simply take an average. Physics demands that we respect the fundamental laws of nature, specifically the Conservation of Internal Energy.

When two non-reacting ideal gases are mixed, the total internal energy of the mixture is simply the sum of the internal energies of the individual gases:

We know that the internal energy of an ideal gas is given by $U=n{C}_{v}T$. Substituting this into our conservation equation, we get:

Since the temperature $T$ is common and cancels out, and knowing that $C_v=\frac{R}{\gamma -1}$, we arrive at our master equation for the mixture:

Now, let's bring in the values from our problem. We have $1\text{ mol}$ of each gas, so $n_1=1$ and $n_2=1$. Their respective gamma values are ${\gamma }_1=\frac{7}{5}$ and ${\gamma }_2=\frac{5}{3}$. Substituting these into our master equation:

Let's simplify the denominators carefully. For the first gas, $\frac{7}{5}-1=\frac{2}{5}$. For the second gas, $\frac{5}{3}-1=\frac{2}{3}$. Plugging these back in:

Inverting the fractions on the right side gives:

Adding the fractions yields $\frac{8}{2}$, which simplifies to $4$. So, we have:

Rearranging to solve for ${\gamma }_{\text{mix}}-1$:

Finally, adding $1$ to both sides gives us our answer:

You might look at the options and panic because $\frac{3}{2}$ is nowhere to be found! But don't rush. Look closely at option (c), which is $\frac{24}{16}$. If you divide the numerator and the denominator by $8$, you get exactly $\frac{3}{2}$. The examiner set a clever little trap to test your confidence in your own calculation. Always trust your math!